Note that the computational complexity of the Ho-Kalman Algorithm in Alg. REF  (deterministic) is dominated by the cost of computing the SVD (Step 7), which is \(O(pT_1\times mT_2\times n)\)  when using the Krylov method (see e.g. [1]}, [2]}). Therefore, we want to use a small \(T\)  to reduce the computational cost. However, to satisfy the second assumption that \(n = \text{rank}(\mathcal {H}) \le \min \lbrace T_1, T_2\rbrace ,\)  where \(T_1+T_2 +1 =T,\)  the smallest \(T\)  we can choose is \(2n+1\)  with \(T_1=T_2=n.\)  In summary, the lowest achievable computational cost for SVD is \(O(n^3).\)  Such dependency on the system dimension is prohibitive for large-scale
systems (e.g. systems with \(n=100\)  as we show in Section ). Motivated by the drawbacks of the existing method,
we aim to answer the following question:
